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If $A= \bigoplus_{i\ge 0}A_i$ is a graded commutative Noetherian algebra over a field, its Poincaré series is given by $P(t) = \sum_{i\ge 0} \dim(A_i)t^i$. Although the definition of $P(t)$ only depends on the graded vector space underlying $A$, the Krull dimension of the ring $A$ can be obtained from the Poincaré series as the order of the pole of $P(t)$ at $t=1$.

Question: Are there other information about the ring structure of $A$ that can be obtained from $P(t)$ ?

Since I'm particularly interested in cases where $A$ is the cohomology ring of a finite group, I'm also looking for an example of finite groups whose mod-p cohomology rings are not isomorphic but have identical Poincaré series.

Thanks in advance.

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up vote 8 down vote accepted

I don't know if it's possible to obtain further information on the ring structure in general. However, if $A$ is the mod-p cohomology ring of a finite group, a result of Benson and Carlson states that if $A$ is Cohen-Macaulay, then $P(t)$ satisfies the functional equation $$P(1/t) = (-1)^d P(t)\hspace{90pt}(\ast)$$ where $d$ is the Krull dimension of $A$. Conversely, if a given $P(t)$ doesn't satisfy this equation, you know that $A$ isn't Cohen-Macaulay, i.e. the depth of $A$ is less than its Krull dimension.

In case $A$ has Krull dimension 2, than $A$ is Cohen-Macaulay iff $P(t)$ satisfies $(\ast)$. These results can be found in the paper Benson, Carlson: Functional equations for Poincaré series in group cohomology. Bull. London Math. Society 26(1994), 438-448.

As an example for groups with identical Poincaré series you can take $E := \mathbb{Z}/2 \times \mathbb{Z}/2$ and $D_8$, the dihedral group of order 8. Their cohomology rings $$H^\ast(E;\mathbb{F}_2)=\mathbb{F}_2[x,y],\;\; |x|=|y|=1$$ $$H^\ast(D_8;\mathbb{F}_2)=\mathbb{F}_2[x,y,z]/(xy),\;\; |x|=|y|=1, |z|=2$$ aren't isomorphic (since only one is a domain) but both have Poincaré series $P(t)=\frac{1}{(1-t)^2}$.


Added: There is a paper of R. Stanley (who is also active on MO) that contains some properties of the Poincaré series that may be of interest. I just quote a few:

  • If $A$ is Gorenstein of Krull dimension $d$, then $P(1/t)=(-1)^dt^aP(t)$ for some integer $a$. This generalizes $(\ast)$ (with a=0) because the mod-p cohomology ring of a finite group is Cohen-Macaulay iff its Gorenstein.

  • $A$ is a complete intersection with generators of degree 1 iff $P(t)$ has the form $$P(t)=\frac{\prod_{i=1}^l(1+t+\cdots+t^{m_i})}{(1-t)^d}$$

  • A sequence of homogeneous elements $x_1,...,x_k \in A$ of positive degrees $n_i$ is regular iff $P(A,t) = P(B,t)/\prod_i (1-t^{n_i})$ where $B=A/(x_1,...,x_n)$.

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In general, I expect there to be statements of the form "nice rings have nice Poincar\'e series", so in the direction of your title, "nasty Poincar\'e series implies nasty ring". –  Allen Knutson Jul 6 '12 at 17:05
    
@Allen: What do you mean by "nice rings" resp. "nice Poincaré series" ? Can you give me some more details on the statements you expect to be there. References are also welcome. Thanks. –  tj_ Jul 6 '12 at 21:57
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